Two-phase almost minimizers for a fractional free boundary problem

In this paper, we study almost minimizers to a fractional Alt-Caffarelli-Friedman type functional. Our main results concern the optimal $C^{0,s}$ regularity of almost minimizers as well as the structure of the free boundary. We first prove that the two free boundaries $F^{+}(u)=\mathrm{\partial }\{u(\cdot ,0)>0\}$ and $F^{-}(u)=\mathrm{\partial }\{u(\cdot ,0)<0\}$ cannot touch, that is, $F^{+}(u)\cap F^{-}(u)=\mathrm{\varnothing }$. Lastly, we prove a flatness implies $C^{1,\gamma }$ result for the free boundary.